Hi mate,
My sister has a summer house at the bottom of her garden that she uses for her Rabbits and guinea pigs. A couple of years ago I put in a solar system connected to an old truck battery that I had lying around.
I went for a 30w solar panel connected to a charge controller (important). The solar panel is mounted on the roof in the direction that the sun is mainly in during summer.
The Lighting is powered directly from the charge controller. The charge controller I used had functions such as timers etc so that the lights could come on automatically at certain times and also it protected the battery from deep discharge and overcharge. There are simply 2 wires that go from the charge controller to the battery (+ and -) which is protected by a 10amp fuse at the battery end. The lighting and other electrical items then simply come of the charge controllers outputs.
During this last summer, when in direct sunlight, the battery was charging at around 2.5 amps for around 10hrs a day. That was more than enough to keep the battery nearly fully charged despite the lights being used for a few hours a night. I also connected up a bank of cigarette lighter sockets that I had lying around and she was able to charge her phone/ ipad/ baby monitor etc whilst out in the garden during the day.
When it was really hot for a week or so, I took a 1000w inverter round and she ran a small tower fan in there to cool it down a bit for them, as she doesnt like leaving the door open at night incase a fox gets in! That ran all night with no problems (although obv a truck battery will have a higher Ah rating than yours probably) and the battery was back up to full by the next evening.
Re which lights to use... Ive done several fit outs on horse box livings and in vans etc, and I would recommend only using LED's nowadays. Try and avoid searching specifically for "caravan" lights as that seems to add 50% to the price! In my sisters case, I used 2 1.5m 12v led strips. One either side. They provide more than enough light and draw very little current.
As you say, as its essentially a 12v system, all the fuses and fuse holders can be 12v blade fuses.
Re the gasses... I wouldnt worry... a shed isnt air tight and is fairly draughty anyway. Also a battery being Charged at around 2/3 amps isnt really going to produce much gas. Never had a problem with my sisters and that has double glazing and air tight door.
Hope that helps,
Alan